If `lambda = `-2, determine the value of `|(0,2lambda,1),(lambda^(2),0,3lambda^(2)+1),(-1,6lambda-1,0)|`.

If the rank of the matrix [(lambda,-1,0),(0, lambda,-1),(-1,0,lambda)] is 2, then find lambda .

Show that if lambda_(1), lambda_(2), ...., lamnda_(n) are n eigenvalues of a square matrix a of order n, then the eigenvalues of the matric A^(2) are lambda_(1)^(2), lambda_(2)^(2),..., lambda_(n)^(2) .

If A = [(lambda,1),(-1,-lambda)] , then for what value of lambda,A^(2)=0 ?

If lambda_(1) and lambda_(2) be two values of lambda for which the expression x^(2)+(2-lambda) x+lambda-(3)/(4) becomes a perfect square, then calculate the value of (lambda_(1)^(2)+lambda_(2)^(2)) .

If [(lambda^(2)-2lambda+1,lambda-2),(1-lambda^(2)+3lambda,1-lambda^(2))]=Alambda^(2)+Blambda+C , where A, B and C are matrices then find matrices B and C.

If |vec a| = 4 and -3 le lambda le 2 then the range of |lambda vec a|

Portion of asymptote of hyperbola x^2/a^2-y^2/b^2 = 1 (between centre and the tangent at vertex) in the first quadrant is cut by the line y + lambda(x-a)=0 (lambda is a parameter) then (A) lambda in R (B) lambda in (0,oo) (C) lambda in (-oo,0) (D) lambda in R-{0}

If |(a^2+lambda^2, a b+clambda, ca-blambda), (a b-clambda, b^2+lambda^2, b c+a), (ca+blambda, bc-alambda, c^2+lambda^2)| |(lambda, c, -b), (-c,lambda, a), (b,-a,lambda)|=(1+a^2+b^2+c^2)^3 , then he value of lambda is a. 8 b. 27 c. 1 d. -1

Find all values of lambda for which the (lambda-1)x+(3lambda+1)y+2lambdaz=0(lambda-1)x+(4lambda-2)y+(lambda+3)z=0 2x+(3lambda+1)y+3(lambda-1)z=0 possess non-trivial solution and find the ratios x:y:z, where lambda has the smallest of these value.

If the equation 4x^2-x-1=0a n d3x^2+(lambda+mu)x+lambda-mu=0 have a root common, then the rational values of lambdaa n dmu are a. lambda=(-3)/4 b. lambda=0 c. mu=3/4 b. mu=0